## Project 3: Multi-Scale Analysis of Phase
Boundaries under
the Influence of Fluctuations and Random Fields

### Participants

Patrick Dondl, Stephan
Luckhaus, **Max
von Renesse**, Michael
Scheutzow
### Summary

We continue our analysis of phase field and sharp interface models for the boundary evolution of two-component systems in the presence of inhomogeneous or random external fields. As before the two reference models of the Mean Curvature Flow (MCF) and its linearization---in the presence of a short-range spatial noise usually called the Quenched Edwards-Wilkinson model - shall be studied. In contrast to the previous research period we will lay emphasis on the dynamics of these models under the influence of a spatially extended and time dependent random forcing. Our main interest will be the long time behaviour of these systems. In particular we want to address the the following questions/phenomena:
Pinning/Depinning behaviour for QEW with random forcing,

Transition to stick-slip bahaviour of QEW in a random environment

Scaling behaviour of stochastic mean curvature flow far from equilibrium

Ergodicity of and effective diffusive dynamics of stochastic mean curvature flow

Besides familiar arguments new mathematical approaches to both MCF and QEW with random forcing will be applied at various stages of our investigation. The methods we want to use are taken from recent progress in stochastic optimal control and stochastic viscosity solutions, maximally monotone stochastic evolution equations and ergodic theory of degenerate SPDEs.

### Some related earlier preprints

Anna De Masi, Stephan Luckhaus and
Errico Presutti.
**Two scale hydrodynamic limit for a model of malignant tumor cells.**

*Annales Henri Poincaré* **43**, 257-297 (2007).

MPI-MIS
Preprint 2/2005.

Anna De Masi, Nicolas Dirr and
Errico Presutti:
**Interface Instability
under Forced Displacements**

*Annales Henri Poincaré* **7:3**, 471-511 (2006)

MPI-MIS
Preprint 5/2005.

Nicolas Dirr and Nung Kwan (Aaron) Yip.

**Pinning
and de-pinning phenomena in front propagation in heterogeneous media.
**

Interfaces and Free Boundaries 8:1
79-109 (2006).

MPI-MIS
Preprint 50/2005.

Nicolas Dirr, Marcello Lucia and
Matteo Novaga.
**Gamma-convergence of the Allen-Cahn energy with
an oscillating forcing term.**

*Interfaces and Free Boundaries* **8:1**, 57-78 (2006).

MPI-MIS
Preprint 38/2005.

Nicolas Dirr and Stephan Luckhaus.
**Mesoscopic limit for non-isothermal phase transition.**

*Markov Process. Relat. Fields* **7:3**, 355-381 (2001).

MPI-MIS
Preprint 70/2001.

### Achievements of the Research Group (funding period 2006/08)

M. v. Renesse and K.-T. Sturm:
**Entropic measure and Wasserstein diffusion,**

*Ann. Probab.*, to appear

preprint

S. Andres and M. v. Renesse:
**Particle approximation of the Wasserstein diffusion,**

preprint

M. v. Renesse, M. Yor and L. Zambotti:
**Quasi-invariance properties of a class of subordinators,**

*Stoch. Proc. Appl.*, to appear

preprint

M. v. Renesse:
**An optimal transport view on Schrödinger's equation**,

preprint

M. v. Renesse:
**On local Poincare via transportation**,

*Math. Z.* **259:1**, 21-31 (2008).

preprint

G. Dimitroff and M. Scheutzow:
**Dispersion of volume under the action of
isotropic Brownian flows.**

*Stoch. Proc. Appl*, to appear.

preprint

M. Scheutzow:
**Chaining techniques and their application to stochastic flows,**

in: Trends in Stochastic Analysis , eds: Blath, J., Mörters, P., Scheutzow, M., Cambridge University Press, to appear.

preprint

Luca Mugnai and Matthias Röger:
**The Allen-Cahn action functional in higher dimensions,**

*Interfaces and Free Boundaries* **10:1**, 45-78, (2008).

Nicolas Dirr, Georgia Karali and Aaron Nung Kwan Yip:
**Pulsating Wave for Mean Curvature Flow in Inhomogeneous Medium,**

*European Journal of Applied Mathematics*, to appear

preprint

Nicolas Dirr, Marcello Lucia and Matteo Novaga:
**Gradient theory of phase transitions with a rapidly
oscillating forcing term,**

*Asymptotic Analysis*, to appear

preprint

Nicolas Dirr and Enza Orlandi:
**Sharp-interface limit of a mesoscopic free energy with
a random external field,**

preprint

Giovanni Bellettini, Anna De Masi, Nicolas Dirr and
Errico Presutti:
**Tunneling in two dimensions,**

*Commun. Math. Phys.* **269**, 715-763 (2007).

offprint,